P.R.-regulated systems of notation and the subrecursive hierarchy equivalence property
Fred Zemke · Transactions of the American Mathematical Society · 1977
We can attempt to extend the Grzegorczyk Hierarchy transfinitely by defining a sequence of functions indexed by the elements of a system of notation S \mathcal {S} , using either iteration (majorization) or enumeration techniques to define the functions. (The hierarchy is then the sequence of classes of functions elementary in the functions of the sequence of functions.) In this paper we consider two sequences { F s } s ∈ S {\{ {F_s}\} _{s \in \mathcal {S}}} and { G s } s ∈ S {\{ {G_s}\} _{s \in \mathcal {S}}} defined by iteration and a sequence { E s } s ∈ S {\{ {E_s}\} _{s \in \mathcal {S}}} defined by enumeration; the corresponding hierarchies are { F s } , { G s } , { E s } \{ {\mathcal {F}_s}\} ,\{ {\mathcal {G}_s}\} ,\{ \mathcal {E}{_s}\} . We say that S \mathcal {S} has the subrecursive hierarchy equivalence property if these two conditions hold: (I)